Machine learning models often inherit biases from historical data, raising critical concerns about fairness and accountability. Conventional fairness interventions typically require access to sensitive attributes like gender or race, but privacy and legal restrictions frequently limit their use. To address this challenge, we propose a framework that infers sensitive attributes from auxiliary features and integrates fairness constraints into model training. Our approach mitigates bias while preserving predictive accuracy, offering a practical solution for fairness-aware learning. Empirical evaluations validate its effectiveness, contributing to the advancement of more equitable algorithmic decision-making.

This paper proposes StrEBM, a structured latent energy-based model for source-wise structured representation learning. The framework is motivated by a broader goal of promoting identifiable and decoupled latent organization by assigning different latent dimensions their own learnable structural biases, rather than constraining the entire latent representation with a single shared energy. In this sense, blind source separation is adopted here as a concrete and verifiable testbed, through which the evolution of latent dimensions toward distinct underlying components can be directly examined. In the proposed framework, latent trajectories are optimized directly together with an observation-generation map and source-wise structural parameters. Each latent dimension is associated with its own energy-based formulation, allowing different latent components to gradually evolve toward distinct source-like roles during training. In the present study, this source-wise energy design is instantiated using Gaussian-process-inspired energies with learnable length-scales, but the framework itself is not restricted to Gaussian processes and is intended as a more general structured latent EBM formulation. Experiments on synthetic multichannel signals under linear and nonlinear mixing settings show that the proposed model can recover source components effectively, providing an initial empirical validation of the framework. At the same time, the study reveals important optimization characteristics, including slow late-stage convergence and reduced stability under nonlinear observation mappings. These findings not only clarify the practical behavior of the current GP-based instantiation, but also establish a basis for future investigation of richer source-wise energy families and more robust nonlinear optimization strategies.

In this paper, we consider the problem of extraction of most informative features from time series that are regarded as observed values of stochastic processes satisfying the It{ô} stochastic differential equations with unknown random drift and diffusion coefficients. We do not attract any additional information and use only the information contained in the time series as it is. Therefore, as additional features, we use the parameters of statistically adjusted mixture-type models of the observed regularities of the behavior of the time series. Several algorithms of construction of these parameters are discussed. These algorithms are based on statistical reconstruction of the coefficients which, in turn, is based on statistical separation of normal mixtures. We obtain two types of parameters by the techniques of the uniform and non-uniform statistical reconstruction of the coefficients of the underlying It{ô} process. The reconstructed coefficients obtained by uniform techniques do not depend on the current value of the process, while the non-uniform techniques reconstruct the coefficients with the account of their dependence on the value of the process. Actually, the non-uniform techniques used in this paper represent a stochastic analog of the Taylor expansion for the time series. The efficiency of the obtained additional features is compared by using them in the autoregressive algorithms of prediction of time series. In order to obtain pure conclusion that is not affected by unwanted factors, say, related to a special choice of the architecture of the neural network prediction methods, we used only simple autoregressive algorithms. We show that the use of additional statistical features improves the prediction.

We develop a geometric framework that links objective accuracy to structural recovery in prototype-based clustering. The analysis is algorithm-agnostic and applies to a broad class of admissible loss functions. We define a clustering condition number that compares within-cluster scale to the minimum loss increase required to move a point across a cluster boundary. When this quantity is small, any solution with a small suboptimality gap must also have a small misclassification error relative to a benchmark partition. The framework also clarifies a fundamental trade-off between robustness and sensitivity to cluster imbalance, leading to sharp phase transitions for exact recovery under different objectives. The guarantees are deterministic and non-asymptotic, and they separate the role of algorithmic accuracy from the intrinsic geometric difficulty of the instance. We further show that errors concentrate near cluster boundaries and that sufficiently deep cluster cores are recovered exactly under strengthened local margins. Together, these results provide a geometric principle for interpreting low objective values as reliable evidence of meaningful clustering structure.

This paper presents a Structured Source-Wise Adaptive Diffusion Framework for linear and nonlinear blind source separation. The framework interprets each latent dimension as a source component and assigns to it an individual adaptive diffusion mechanism, thereby establishing source-wise latent modeling rather than relying on a single shared latent prior. The resulting formulation learns source recovery and the mixing/reconstruction process jointly within a unified end-to-end objective, allowing model parameters and latent sources to adapt simultaneously during training. This yields a common framework for both linear and nonlinear blind source separation. In the present instantiation, each source is further equipped with its own adaptive Gaussian process (GP) prior to impose source-wise temporal structure on the latent trajectories, while the overall framework is not restricted to Gaussian process priors and can in principle accommodate other structured source priors. The proposed model thus provides a general structured diffusion-based route to unsupervised source recovery, with potential relevance beyond blind source separation to interpretable latent modeling, source-wise disentanglement, and potentially identifiable nonlinear latent-variable learning under appropriate structural conditions.

We study language models as evolving model organisms and ask when autoregressive next-token learning selects for world-tracking representations. For any encoding of latent world states, the Bayes-optimal next-token cross-entropy decomposes into the irreducible conditional entropy plus a Jensen--Shannon excess term. That excess vanishes if and only if the encoding preserves the training ecology's equivalence classes. This yields a precise notion of ecological veridicality for language models and identifies the minimum-complexity zero-excess solution as the quotient partition by training equivalence. We then determine when this fixed-encoding analysis applies to transformer families: frozen dense and frozen Mixture-of-Experts transformers satisfy it, in-context learning does not enlarge the model's separation set, and per-task adaptation breaks the premise. The framework predicts two characteristic failure modes: simplicity pressure preferentially removes low-gain distinctions, and training-optimal models can still incur positive excess on deployment ecologies that refine the training ecology. A conditional dynamic extension shows how inter-model selection and post-training can recover such gap distinctions under explicit heredity, variation, and selection assumptions. Exact finite-ecology checks and controlled microgpt experiments validate the static decomposition, split-merge threshold, off-ecology failure pattern, and two-ecology rescue mechanism in a regime where the relevant quantities are directly observable. The goal is not to model frontier systems at scale, but to use small language models as laboratory organisms for theory about representational selection.

Machine learning (ML) models have achieved strikingly high accuracies in spectroscopic classification tasks, often without a clear proof that those models used chemically meaningful features. Existing studies have linked these results to data preprocessing choices, noise sensitivity, and model complexity, but no unifying explanation is available so far. In this work, we show that these phenomena arise naturally from the intrinsic high dimensionality of spectral data. Using a theoretical analysis grounded in the Feldman-Hajek theorem and the concentration of measure, we show that even infinitesimal distributional differences, caused by noise, normalisation, or instrumental artefacts, may become perfectly separable in high-dimensional spaces. Through a series of specific experiments on synthetic and real fluorescence spectra, we illustrate how models can achieve near-perfect accuracy even when chemical distinctions are absent, and why feature-importance maps may highlight spectrally irrelevant regions. We provide a rigorous theoretical framework, confirm the effect experimentally, and conclude with practical recommendations for building and interpreting ML models in spectroscopy.

Can a safety gate permit unbounded beneficial self-modification while maintaining bounded cumulative risk? We formalize this question through dual conditions -- requiring sum delta_n < infinity (bounded risk) and sum TPR_n = infinity (unbounded utility) -- and establish a theory of their (in)compatibility. Classification impossibility (Theorem 1): For power-law risk schedules delta_n = O(n^{-p}) with p > 1, any classifier-based gate under overlapping safe/unsafe distributions satisfies TPR_n <= C_alpha * delta_n^beta via Holder's inequality, forcing sum TPR_n < infinity. This impossibility is exponent-optimal (Theorem 3). A second independent proof via the NP counting method (Theorem 4) yields a 13% tighter bound without Holder's inequality. Universal finite-horizon ceiling (Theorem 5): For any summable risk schedule, the exact maximum achievable classifier utility is U*(N, B) = N * TPR_NP(B/N), growing as exp(O(sqrt(log N))) -- subpolynomial. At N = 10^6 with budget B = 1.0, a classifier extracts at most U* ~ 87 versus a verifier's ~500,000. Verification escape (Theorem 2): A Lipschitz ball verifier achieves delta = 0 with TPR > 0, escaping the impossibility. Formal Lipschitz bounds for pre-LayerNorm transformers under LoRA enable LLM-scale verification. The separation is strict. We validate on GPT-2 (d_LoRA = 147,456): conditional delta = 0 with TPR = 0.352. Comprehensive empirical validation is in the companion paper [D2].

Can classifier-based safety gates maintain reliable oversight as AI systems improve over hundreds of iterations? We provide comprehensive empirical evidence that they cannot. On a self-improving neural controller (d=240), eighteen classifier configurations -- spanning MLPs, SVMs, random forests, k-NN, Bayesian classifiers, and deep networks -- all fail the dual conditions for safe self-improvement. Three safe RL baselines (CPO, Lyapunov, safety shielding) also fail. Results extend to MuJoCo benchmarks (Reacher-v4 d=496, Swimmer-v4 d=1408, HalfCheetah-v4 d=1824). At controlled distribution separations up to delta_s=2.0, all classifiers still fail -- including the NP-optimal test and MLPs with 100% training accuracy -- demonstrating structural impossibility. We then show the impossibility is specific to classification, not to safe self-improvement itself. A Lipschitz ball verifier achieves zero false accepts across dimensions d in {84, 240, 768, 2688, 5760, 9984, 17408} using provable analytical bounds (unconditional delta=0). Ball chaining enables unbounded parameter-space traversal: on MuJoCo Reacher-v4, 10 chains yield +4.31 reward improvement with delta=0; on Qwen2.5-7B-Instruct during LoRA fine-tuning, 42 chain transitions traverse 234x the single-ball radius with zero safety violations across 200 steps. A 50-prompt oracle confirms oracle-agnosticity. Compositional per-group verification enables radii up to 37x larger than full-network balls. At d<=17408, delta=0 is unconditional; at LLM scale, conditional on estimated Lipschitz constants.

We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.